Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,19)(13,25)(18,34)(20,29)(21,28)(23,33)(24,32)(26,31)(27,36)(30,35);(2,4)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,28)(15,17)(16,29)(18,25)(22,34)(24,36)(26,35)(27,31)(32,33);(5,8)(6,13)(7,18)(9,25)(10,24)(11,26)(12,27)(14,22)(15,31)(16,32)(17,33)(19,23)(20,35)(21,36)(28,30)(29,34);(5,6)(7,12)(8,13)(9,15)(10,14)(11,16)(17,20)(18,27)(19,28)(21,29)(22,24)(23,30)(25,31)(26,32)(33,35)(34,36)Polycyclic presentation:
2^9|a^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72249958, #72257156, #72257160, … |
| maximal normal subgroup of | #72249958, #72257156, #72257160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302613);